Multiple Solutions For The Biharmonic Equation With Critical Exponent

In this paper, we consider the existence of multiple solutions for the biharmonic equations boundary value problemwhereΩ(?) Rⁿ is a bounded open domain with smooth boundaryλ> 0 is a given constant, p = (N+4)/(N-4) is the critical Sobolev exponent for the embeding H₀²(Ω) (?) L^P+1(Ω) and H₀²(Ω) is a standard Sobolev space with equivalent norm ||Δu||_L²(Ω),Δ²=ΔΔdenotes iterated N-domensional Laplacian.We are interested in problem (1) because of the lack of compactness of corresponding energy functional,for u∈H₀²(Ω), and this leads to many interesting existence and nonex-istence phenomena. It is well known that problem (1) has no nontrival solution ifλ< 0 andΩis a star-shiped domain (see [9]). Forλ> 0, we can prove that problem (1) possesses at least one nontrival solution ifN≥8,λ∈(0,λ₁(N)). Here,λ₁(N) denotes the first eigenvalue ofΔ² in H₀²(Ω). For N = 5,6, 7 we can find two constantsλ**(N) andλ*(N) satiafy-ing0 <λ^**(N)≤λ^*(N) <λ₁(N)such that problem (1) poessesses at least one nontrival solution ifλ∈(λ^*(N),λ₁(N)) and has, in the ball, no nontrival radial solution ifλ∈(0,λ^**(N) (see[8]).We say N is a critical dimension with respect to boundary value problem, if there is a positive constantΛ> 0 such that the necessary condition for the existence of nontrival radial solution to problem (2) withΩ= B_r(0) isλ>Λ, where m∈N, N > 2m,λ∈R¹ and s =(N+2m)/(N-2m) is the critical Sobolev exponent.Many authors has recently investigated the semilinear polyharmonic eigenvalue problem (2). Due to the criticality of exponents, a nontrival solution to problem (2) withΩ= B_R(0) may exist at most forλ≥0, and it also depends on the dimension N. The case m=1 has been studied by Breize and Nirengberg [2]. Pucci and Serrin in their celebrated paper [10] conjectured that the critical dimensions for boundary value problem (2) withΩ=B_R(0) are precisely N = 2m + 1,...,4m - 1.The main idea of this paper is as the following. Firstly we prove the existence of the second nontrival solution for problem (1) by the solution sequenece in subcritical case. Secondly we assume thatΩ=B_R(0), then we get the existence of the positive and sign-changing solutions by Green function .